3frac (problem 3.3.3)

Percentage Accurate: 69.8% → 99.5%

Time: 9.6s

Alternatives: 12

Speedup: 1.0×

Specification

\[\left|x\right| > 1\]

Math

\[\begin{array}{l} \\ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))

C

double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 / (x + 1.0d0)) - (2.0d0 / x)) + (1.0d0 / (x - 1.0d0))
end function

Java

public static double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

Python

def code(x):
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0))

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x - 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
end

Wolfram

code[x_] := N[(N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 69.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))

C

double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 / (x + 1.0d0)) - (2.0d0 / x)) + (1.0d0 / (x - 1.0d0))
end function

Java

public static double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

Python

def code(x):
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0))

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x - 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
end

Wolfram

code[x_] := N[(N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \left(2 + \frac{2}{x \cdot x}\right) \cdot {x}^{-3} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (+ 2.0 (/ 2.0 (* x x))) (pow x -3.0)))

C

double code(double x) {
	return (2.0 + (2.0 / (x * x))) * pow(x, -3.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (2.0d0 + (2.0d0 / (x * x))) * (x ** (-3.0d0))
end function

Java

public static double code(double x) {
	return (2.0 + (2.0 / (x * x))) * Math.pow(x, -3.0);
}

Python

def code(x):
	return (2.0 + (2.0 / (x * x))) * math.pow(x, -3.0)

Julia

function code(x)
	return Float64(Float64(2.0 + Float64(2.0 / Float64(x * x))) * (x ^ -3.0))
end

MATLAB

function tmp = code(x)
	tmp = (2.0 + (2.0 / (x * x))) * (x ^ -3.0);
end

Wolfram

code[x_] := N[(N[(2.0 + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.7%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

3. Simplified 70.7%

   \[\leadsto \color{blue}{\frac{1}{1 + x} + \left(\frac{-2}{x} + \frac{1}{x + -1}\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 98.8%

   \[\leadsto \color{blue}{\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{3}}} \]
6. Step-by-step derivation

   1. associate-*r/ 98.8%

      \[\leadsto \frac{2 + \color{blue}{\frac{2 \cdot 1}{{x}^{2}}}}{{x}^{3}} \]

   2. metadata-eval 98.8%

      \[\leadsto \frac{2 + \frac{\color{blue}{2}}{{x}^{2}}}{{x}^{3}} \]

7. Simplified 98.8%

   \[\leadsto \color{blue}{\frac{2 + \frac{2}{{x}^{2}}}{{x}^{3}}} \]
8. Step-by-step derivation

   1. div-inv 98.8%

      \[\leadsto \color{blue}{\left(2 + \frac{2}{{x}^{2}}\right) \cdot \frac{1}{{x}^{3}}} \]

   2. +-commutative 98.8%

      \[\leadsto \color{blue}{\left(\frac{2}{{x}^{2}} + 2\right)} \cdot \frac{1}{{x}^{3}} \]

   3. div-inv 98.8%

      \[\leadsto \left(\color{blue}{2 \cdot \frac{1}{{x}^{2}}} + 2\right) \cdot \frac{1}{{x}^{3}} \]

   4. fma-define 98.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{1}{{x}^{2}}, 2\right)} \cdot \frac{1}{{x}^{3}} \]

   5. pow-flip 98.8%

      \[\leadsto \mathsf{fma}\left(2, \color{blue}{{x}^{\left(-2\right)}}, 2\right) \cdot \frac{1}{{x}^{3}} \]

   6. metadata-eval 98.8%

      \[\leadsto \mathsf{fma}\left(2, {x}^{\color{blue}{-2}}, 2\right) \cdot \frac{1}{{x}^{3}} \]

   7. pow-flip 99.5%

      \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, 2\right) \cdot \color{blue}{{x}^{\left(-3\right)}} \]

   8. metadata-eval 99.5%

      \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, 2\right) \cdot {x}^{\color{blue}{-3}} \]

9. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, {x}^{-2}, 2\right) \cdot {x}^{-3}} \]
10. Step-by-step derivation

    1. metadata-eval 99.5%

       \[\leadsto \mathsf{fma}\left(2, {x}^{\color{blue}{\left(-2\right)}}, 2\right) \cdot {x}^{-3} \]

    2. pow-flip 99.5%

       \[\leadsto \mathsf{fma}\left(2, \color{blue}{\frac{1}{{x}^{2}}}, 2\right) \cdot {x}^{-3} \]

    3. pow2 99.5%

       \[\leadsto \mathsf{fma}\left(2, \frac{1}{\color{blue}{x \cdot x}}, 2\right) \cdot {x}^{-3} \]

    4. fma-define 99.5%

       \[\leadsto \color{blue}{\left(2 \cdot \frac{1}{x \cdot x} + 2\right)} \cdot {x}^{-3} \]

    5. div-inv 99.5%

       \[\leadsto \left(\color{blue}{\frac{2}{x \cdot x}} + 2\right) \cdot {x}^{-3} \]

    6. pow2 99.5%

       \[\leadsto \left(\frac{2}{\color{blue}{{x}^{2}}} + 2\right) \cdot {x}^{-3} \]

11. Applied egg-rr 99.5%

    \[\leadsto \color{blue}{\left(\frac{2}{{x}^{2}} + 2\right)} \cdot {x}^{-3} \]
12. Step-by-step derivation

    1. unpow2 99.5%

       \[\leadsto \left(\frac{2}{\color{blue}{x \cdot x}} + 2\right) \cdot {x}^{-3} \]

13. Applied egg-rr 99.5%

    \[\leadsto \left(\frac{2}{\color{blue}{x \cdot x}} + 2\right) \cdot {x}^{-3} \]

14. Final simplification 99.5%

    \[\leadsto \left(2 + \frac{2}{x \cdot x}\right) \cdot {x}^{-3} \]
15. Add Preprocessing

Alternative 2: 99.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot {x}^{-3} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 2.0 (pow x -3.0)))

C

double code(double x) {
	return 2.0 * pow(x, -3.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 * (x ** (-3.0d0))
end function

Java

public static double code(double x) {
	return 2.0 * Math.pow(x, -3.0);
}

Python

def code(x):
	return 2.0 * math.pow(x, -3.0)

Julia

function code(x)
	return Float64(2.0 * (x ^ -3.0))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * (x ^ -3.0);
end

Wolfram

code[x_] := N[(2.0 * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.7%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

3. Simplified 70.7%

   \[\leadsto \color{blue}{\frac{1}{1 + x} + \left(\frac{-2}{x} + \frac{1}{x + -1}\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 98.8%

   \[\leadsto \color{blue}{\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{3}}} \]
6. Step-by-step derivation

   1. associate-*r/ 98.8%

      \[\leadsto \frac{2 + \color{blue}{\frac{2 \cdot 1}{{x}^{2}}}}{{x}^{3}} \]

   2. metadata-eval 98.8%

      \[\leadsto \frac{2 + \frac{\color{blue}{2}}{{x}^{2}}}{{x}^{3}} \]

7. Simplified 98.8%

   \[\leadsto \color{blue}{\frac{2 + \frac{2}{{x}^{2}}}{{x}^{3}}} \]
8. Step-by-step derivation

   1. div-inv 98.8%

      \[\leadsto \color{blue}{\left(2 + \frac{2}{{x}^{2}}\right) \cdot \frac{1}{{x}^{3}}} \]

   2. +-commutative 98.8%

      \[\leadsto \color{blue}{\left(\frac{2}{{x}^{2}} + 2\right)} \cdot \frac{1}{{x}^{3}} \]

   3. div-inv 98.8%

      \[\leadsto \left(\color{blue}{2 \cdot \frac{1}{{x}^{2}}} + 2\right) \cdot \frac{1}{{x}^{3}} \]

   4. fma-define 98.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{1}{{x}^{2}}, 2\right)} \cdot \frac{1}{{x}^{3}} \]

   5. pow-flip 98.8%

      \[\leadsto \mathsf{fma}\left(2, \color{blue}{{x}^{\left(-2\right)}}, 2\right) \cdot \frac{1}{{x}^{3}} \]

   6. metadata-eval 98.8%

      \[\leadsto \mathsf{fma}\left(2, {x}^{\color{blue}{-2}}, 2\right) \cdot \frac{1}{{x}^{3}} \]

   7. pow-flip 99.5%

      \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, 2\right) \cdot \color{blue}{{x}^{\left(-3\right)}} \]

   8. metadata-eval 99.5%

      \[\leadsto \mathsf{fma}\left(2, {x}^{-2}, 2\right) \cdot {x}^{\color{blue}{-3}} \]

9. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, {x}^{-2}, 2\right) \cdot {x}^{-3}} \]

10. Taylor expanded in x around inf 99.1%

    \[\leadsto \color{blue}{2} \cdot {x}^{-3} \]
11. Add Preprocessing

Alternative 3: 69.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x + 1} + \left(\frac{-2}{x} + \frac{1}{x \cdot \left(1 - \frac{1}{x}\right)}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (/ 1.0 (+ x 1.0)) (+ (/ -2.0 x) (/ 1.0 (* x (- 1.0 (/ 1.0 x)))))))

C

double code(double x) {
	return (1.0 / (x + 1.0)) + ((-2.0 / x) + (1.0 / (x * (1.0 - (1.0 / x)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / (x + 1.0d0)) + (((-2.0d0) / x) + (1.0d0 / (x * (1.0d0 - (1.0d0 / x)))))
end function

Java

public static double code(double x) {
	return (1.0 / (x + 1.0)) + ((-2.0 / x) + (1.0 / (x * (1.0 - (1.0 / x)))));
}

Python

def code(x):
	return (1.0 / (x + 1.0)) + ((-2.0 / x) + (1.0 / (x * (1.0 - (1.0 / x)))))

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(x + 1.0)) + Float64(Float64(-2.0 / x) + Float64(1.0 / Float64(x * Float64(1.0 - Float64(1.0 / x))))))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / (x + 1.0)) + ((-2.0 / x) + (1.0 / (x * (1.0 - (1.0 / x)))));
end

Wolfram

code[x_] := N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 / x), $MachinePrecision] + N[(1.0 / N[(x * N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.7%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

3. Simplified 70.7%

   \[\leadsto \color{blue}{\frac{1}{1 + x} + \left(\frac{-2}{x} + \frac{1}{x + -1}\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 70.7%

   \[\leadsto \frac{1}{1 + x} + \left(\frac{-2}{x} + \frac{1}{\color{blue}{x \cdot \left(1 - \frac{1}{x}\right)}}\right) \]

6. Final simplification 70.7%

   \[\leadsto \frac{1}{x + 1} + \left(\frac{-2}{x} + \frac{1}{x \cdot \left(1 - \frac{1}{x}\right)}\right) \]
7. Add Preprocessing

Alternative 4: 69.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x + 1} + \left(\frac{-2}{x} + \frac{1}{x + -1}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (/ 1.0 (+ x 1.0)) (+ (/ -2.0 x) (/ 1.0 (+ x -1.0)))))

C

double code(double x) {
	return (1.0 / (x + 1.0)) + ((-2.0 / x) + (1.0 / (x + -1.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / (x + 1.0d0)) + (((-2.0d0) / x) + (1.0d0 / (x + (-1.0d0))))
end function

Java

public static double code(double x) {
	return (1.0 / (x + 1.0)) + ((-2.0 / x) + (1.0 / (x + -1.0)));
}

Python

def code(x):
	return (1.0 / (x + 1.0)) + ((-2.0 / x) + (1.0 / (x + -1.0)))

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(x + 1.0)) + Float64(Float64(-2.0 / x) + Float64(1.0 / Float64(x + -1.0))))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / (x + 1.0)) + ((-2.0 / x) + (1.0 / (x + -1.0)));
end

Wolfram

code[x_] := N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 / x), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.7%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

3. Simplified 70.7%

   \[\leadsto \color{blue}{\frac{1}{1 + x} + \left(\frac{-2}{x} + \frac{1}{x + -1}\right)} \]
4. Add Preprocessing

5. Final simplification 70.7%

   \[\leadsto \frac{1}{x + 1} + \left(\frac{-2}{x} + \frac{1}{x + -1}\right) \]
6. Add Preprocessing

Alternative 5: 69.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x + -1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (/ 1.0 (+ x -1.0)) (- (/ 1.0 (+ x 1.0)) (/ 2.0 x))))

C

double code(double x) {
	return (1.0 / (x + -1.0)) + ((1.0 / (x + 1.0)) - (2.0 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / (x + (-1.0d0))) + ((1.0d0 / (x + 1.0d0)) - (2.0d0 / x))
end function

Java

public static double code(double x) {
	return (1.0 / (x + -1.0)) + ((1.0 / (x + 1.0)) - (2.0 / x));
}

Python

def code(x):
	return (1.0 / (x + -1.0)) + ((1.0 / (x + 1.0)) - (2.0 / x))

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(x + -1.0)) + Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(2.0 / x)))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / (x + -1.0)) + ((1.0 / (x + 1.0)) - (2.0 / x));
end

Wolfram

code[x_] := N[(N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.7%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
2. Add Preprocessing

3. Final simplification 70.7%

   \[\leadsto \frac{1}{x + -1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right) \]
4. Add Preprocessing

Alternative 6: 68.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x} \cdot \left(1 + \left(\frac{1}{x} + -1\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (/ 1.0 x) (+ 1.0 (+ (/ 1.0 x) -1.0))))

C

double code(double x) {
	return (1.0 / x) * (1.0 + ((1.0 / x) + -1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / x) * (1.0d0 + ((1.0d0 / x) + (-1.0d0)))
end function

Java

public static double code(double x) {
	return (1.0 / x) * (1.0 + ((1.0 / x) + -1.0));
}

Python

def code(x):
	return (1.0 / x) * (1.0 + ((1.0 / x) + -1.0))

Julia

function code(x)
	return Float64(Float64(1.0 / x) * Float64(1.0 + Float64(Float64(1.0 / x) + -1.0)))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / x) * (1.0 + ((1.0 / x) + -1.0));
end

Wolfram

code[x_] := N[(N[(1.0 / x), $MachinePrecision] * N[(1.0 + N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.7%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

3. Simplified 70.7%

   \[\leadsto \color{blue}{\frac{1}{1 + x} + \left(\frac{-2}{x} + \frac{1}{x + -1}\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 69.6%

   \[\leadsto \color{blue}{\frac{1}{x}} + \left(\frac{-2}{x} + \frac{1}{x + -1}\right) \]

6. Taylor expanded in x around inf 69.6%

   \[\leadsto \frac{1}{x} + \color{blue}{\frac{\frac{1}{x} - 1}{x}} \]
7. Step-by-step derivation

   1. +-commutative 69.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{x} - 1}{x} + \frac{1}{x}} \]

   2. div-inv 69.6%

      \[\leadsto \color{blue}{\left(\frac{1}{x} - 1\right) \cdot \frac{1}{x}} + \frac{1}{x} \]

   3. fma-define 69.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{x} - 1, \frac{1}{x}, \frac{1}{x}\right)} \]

   4. sub-neg 69.6%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{x} + \left(-1\right)}, \frac{1}{x}, \frac{1}{x}\right) \]

   5. metadata-eval 69.6%

      \[\leadsto \mathsf{fma}\left(\frac{1}{x} + \color{blue}{-1}, \frac{1}{x}, \frac{1}{x}\right) \]

8. Applied egg-rr 69.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{x} + -1, \frac{1}{x}, \frac{1}{x}\right)} \]
9. Step-by-step derivation

   1. fma-undefine 69.6%

      \[\leadsto \color{blue}{\left(\frac{1}{x} + -1\right) \cdot \frac{1}{x} + \frac{1}{x}} \]

   2. *-commutative 69.6%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(\frac{1}{x} + -1\right)} + \frac{1}{x} \]

   3. *-rgt-identity 69.6%

      \[\leadsto \frac{1}{x} \cdot \left(\frac{1}{x} + -1\right) + \color{blue}{\frac{1}{x} \cdot 1} \]

   4. distribute-lft-out 69.6%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(\left(\frac{1}{x} + -1\right) + 1\right)} \]

10. Simplified 69.6%

    \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(\left(\frac{1}{x} + -1\right) + 1\right)} \]

11. Final simplification 69.6%

    \[\leadsto \frac{1}{x} \cdot \left(1 + \left(\frac{1}{x} + -1\right)\right) \]
12. Add Preprocessing

Alternative 7: 68.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x + -1} + \frac{-1}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (/ 1.0 (+ x -1.0)) (/ -1.0 x)))

C

double code(double x) {
	return (1.0 / (x + -1.0)) + (-1.0 / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / (x + (-1.0d0))) + ((-1.0d0) / x)
end function

Java

public static double code(double x) {
	return (1.0 / (x + -1.0)) + (-1.0 / x);
}

Python

def code(x):
	return (1.0 / (x + -1.0)) + (-1.0 / x)

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(x + -1.0)) + Float64(-1.0 / x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / (x + -1.0)) + (-1.0 / x);
end

Wolfram

code[x_] := N[(N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.7%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
2. Step-by-step derivation

   1. +-commutative 70.7%

      \[\leadsto \color{blue}{\frac{1}{x - 1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right)} \]

   2. associate-+r- 70.7%

      \[\leadsto \color{blue}{\left(\frac{1}{x - 1} + \frac{1}{x + 1}\right) - \frac{2}{x}} \]

   3. sub-neg 70.7%

      \[\leadsto \color{blue}{\left(\frac{1}{x - 1} + \frac{1}{x + 1}\right) + \left(-\frac{2}{x}\right)} \]

   4. remove-double-neg 70.7%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{\left(-\left(-x\right)\right)} + 1}\right) + \left(-\frac{2}{x}\right) \]

   5. neg-sub0 70.7%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{\left(0 - \left(-x\right)\right)} + 1}\right) + \left(-\frac{2}{x}\right) \]

   6. associate-+l- 70.7%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{0 - \left(\left(-x\right) - 1\right)}}\right) + \left(-\frac{2}{x}\right) \]

   7. neg-sub0 70.7%

      \[\leadsto \left(\frac{1}{x - 1} + \frac{1}{\color{blue}{-\left(\left(-x\right) - 1\right)}}\right) + \left(-\frac{2}{x}\right) \]

   8. distribute-neg-frac2 70.7%

      \[\leadsto \left(\frac{1}{x - 1} + \color{blue}{\left(-\frac{1}{\left(-x\right) - 1}\right)}\right) + \left(-\frac{2}{x}\right) \]

   9. distribute-frac-neg2 70.7%

      \[\leadsto \left(\frac{1}{x - 1} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) + \color{blue}{\frac{2}{-x}} \]

   10. associate-+r+ 70.7%

       \[\leadsto \color{blue}{\frac{1}{x - 1} + \left(\left(-\frac{1}{\left(-x\right) - 1}\right) + \frac{2}{-x}\right)} \]

   11. +-commutative 70.7%

       \[\leadsto \frac{1}{x - 1} + \color{blue}{\left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right)} \]

   12. remove-double-neg 70.7%

       \[\leadsto \color{blue}{\left(-\left(-\frac{1}{x - 1}\right)\right)} + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   13. distribute-neg-frac2 70.7%

       \[\leadsto \left(-\color{blue}{\frac{1}{-\left(x - 1\right)}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   14. sub0-neg 70.7%

       \[\leadsto \left(-\frac{1}{\color{blue}{0 - \left(x - 1\right)}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   15. associate-+l- 70.7%

       \[\leadsto \left(-\frac{1}{\color{blue}{\left(0 - x\right) + 1}}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

   16. neg-sub0 70.7%

       \[\leadsto \left(-\frac{1}{\color{blue}{\left(-x\right)} + 1}\right) + \left(\frac{2}{-x} + \left(-\frac{1}{\left(-x\right) - 1}\right)\right) \]

3. Simplified 70.7%

   \[\leadsto \color{blue}{\frac{1}{x + -1} + \left(\frac{-2}{x} - \frac{1}{-1 - x}\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 69.8%

   \[\leadsto \frac{1}{x + -1} + \color{blue}{-1 \cdot \frac{1 + \frac{1}{x}}{x}} \]
6. Step-by-step derivation

   1. associate-*r/ 69.8%

      \[\leadsto \frac{1}{x + -1} + \color{blue}{\frac{-1 \cdot \left(1 + \frac{1}{x}\right)}{x}} \]

   2. neg-mul-1 69.8%

      \[\leadsto \frac{1}{x + -1} + \frac{\color{blue}{-\left(1 + \frac{1}{x}\right)}}{x} \]

   3. distribute-neg-in 69.8%

      \[\leadsto \frac{1}{x + -1} + \frac{\color{blue}{\left(-1\right) + \left(-\frac{1}{x}\right)}}{x} \]

   4. metadata-eval 69.8%

      \[\leadsto \frac{1}{x + -1} + \frac{\color{blue}{-1} + \left(-\frac{1}{x}\right)}{x} \]

   5. distribute-neg-frac 69.8%

      \[\leadsto \frac{1}{x + -1} + \frac{-1 + \color{blue}{\frac{-1}{x}}}{x} \]

   6. metadata-eval 69.8%

      \[\leadsto \frac{1}{x + -1} + \frac{-1 + \frac{\color{blue}{-1}}{x}}{x} \]

7. Simplified 69.8%

   \[\leadsto \frac{1}{x + -1} + \color{blue}{\frac{-1 + \frac{-1}{x}}{x}} \]

8. Taylor expanded in x around inf 69.6%

   \[\leadsto \frac{1}{x + -1} + \color{blue}{\frac{-1}{x}} \]
9. Add Preprocessing

Alternative 8: 68.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ -1 + \left(1 + \frac{1}{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ -1.0 (+ 1.0 (/ 1.0 x))))

C

double code(double x) {
	return -1.0 + (1.0 + (1.0 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-1.0d0) + (1.0d0 + (1.0d0 / x))
end function

Java

public static double code(double x) {
	return -1.0 + (1.0 + (1.0 / x));
}

Python

def code(x):
	return -1.0 + (1.0 + (1.0 / x))

Julia

function code(x)
	return Float64(-1.0 + Float64(1.0 + Float64(1.0 / x)))
end

MATLAB

function tmp = code(x)
	tmp = -1.0 + (1.0 + (1.0 / x));
end

Wolfram

code[x_] := N[(-1.0 + N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.7%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

3. Simplified 70.7%

   \[\leadsto \color{blue}{\frac{1}{1 + x} + \left(\frac{-2}{x} + \frac{1}{x + -1}\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 69.6%

   \[\leadsto \color{blue}{\frac{1}{x}} + \left(\frac{-2}{x} + \frac{1}{x + -1}\right) \]

6. Taylor expanded in x around 0 5.3%

   \[\leadsto \color{blue}{\frac{-1}{x}} \]
7. Step-by-step derivation

   1. div-inv 5.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{1}{x}} \]

   2. pow1 5.3%

      \[\leadsto -1 \cdot \frac{1}{\color{blue}{{x}^{1}}} \]

   3. metadata-eval 5.3%

      \[\leadsto -1 \cdot \frac{1}{{x}^{\color{blue}{\left(\frac{2}{2}\right)}}} \]

   4. sqrt-pow1 57.5%

      \[\leadsto -1 \cdot \frac{1}{\color{blue}{\sqrt{{x}^{2}}}} \]

   5. pow2 57.5%

      \[\leadsto -1 \cdot \frac{1}{\sqrt{\color{blue}{x \cdot x}}} \]

   6. sqr-neg 57.5%

      \[\leadsto -1 \cdot \frac{1}{\sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}} \]

   7. sqrt-unprod 3.7%

      \[\leadsto -1 \cdot \frac{1}{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}} \]

   8. add-sqr-sqrt 6.6%

      \[\leadsto -1 \cdot \frac{1}{\color{blue}{-x}} \]

   9. metadata-eval 6.6%

      \[\leadsto -1 \cdot \frac{\color{blue}{--1}}{-x} \]

   10. frac-2neg 6.6%

       \[\leadsto -1 \cdot \color{blue}{\frac{-1}{x}} \]

8. Applied egg-rr 6.6%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1}{x}} \]
9. Step-by-step derivation

   1. associate-*r/ 6.6%

      \[\leadsto \color{blue}{\frac{-1 \cdot -1}{x}} \]

   2. metadata-eval 6.6%

      \[\leadsto \frac{\color{blue}{1}}{x} \]

10. Simplified 6.6%

    \[\leadsto \color{blue}{\frac{1}{x}} \]
11. Step-by-step derivation

    1. pow1 6.6%

       \[\leadsto \frac{1}{\color{blue}{{x}^{1}}} \]

    2. metadata-eval 6.6%

       \[\leadsto \frac{1}{{x}^{\color{blue}{\left(\frac{2}{2}\right)}}} \]

    3. sqrt-pow1 57.4%

       \[\leadsto \frac{1}{\color{blue}{\sqrt{{x}^{2}}}} \]

    4. unpow2 57.4%

       \[\leadsto \frac{1}{\sqrt{\color{blue}{x \cdot x}}} \]

    5. sqr-neg 57.4%

       \[\leadsto \frac{1}{\sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}} \]

    6. sqrt-unprod 3.0%

       \[\leadsto \frac{1}{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}} \]

    7. add-sqr-sqrt 5.3%

       \[\leadsto \frac{1}{\color{blue}{-x}} \]

    8. expm1-log1p-u 5.3%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{-x}\right)\right)} \]

    9. metadata-eval 5.3%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{--1}}{-x}\right)\right) \]

    10. frac-2neg 5.3%

        \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{-1}{x}}\right)\right) \]

    11. log1p-define 69.3%

        \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{-1}{x}\right)}\right) \]

    12. +-commutative 69.3%

        \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left(\frac{-1}{x} + 1\right)}\right) \]

    13. expm1-undefine 69.3%

        \[\leadsto \color{blue}{e^{\log \left(\frac{-1}{x} + 1\right)} - 1} \]

    14. add-exp-log 69.3%

        \[\leadsto \color{blue}{\left(\frac{-1}{x} + 1\right)} - 1 \]

    15. +-commutative 69.3%

        \[\leadsto \color{blue}{\left(1 + \frac{-1}{x}\right)} - 1 \]

    16. frac-2neg 69.3%

        \[\leadsto \left(1 + \color{blue}{\frac{--1}{-x}}\right) - 1 \]

    17. metadata-eval 69.3%

        \[\leadsto \left(1 + \frac{\color{blue}{1}}{-x}\right) - 1 \]

    18. add-sqr-sqrt 37.4%

        \[\leadsto \left(1 + \frac{1}{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}\right) - 1 \]

    19. sqrt-unprod 69.5%

        \[\leadsto \left(1 + \frac{1}{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}}\right) - 1 \]

    20. sqr-neg 69.5%

        \[\leadsto \left(1 + \frac{1}{\sqrt{\color{blue}{x \cdot x}}}\right) - 1 \]

    21. sqrt-prod 32.1%

        \[\leadsto \left(1 + \frac{1}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right) - 1 \]

    22. add-sqr-sqrt 69.8%

        \[\leadsto \left(1 + \frac{1}{\color{blue}{x}}\right) - 1 \]

12. Applied egg-rr 69.8%

    \[\leadsto \color{blue}{\left(1 + \frac{1}{x}\right) - 1} \]

13. Final simplification 69.8%

    \[\leadsto -1 + \left(1 + \frac{1}{x}\right) \]
14. Add Preprocessing

Alternative 9: 68.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x} - \frac{1}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ 1.0 x) (/ 1.0 x)))

C

double code(double x) {
	return (1.0 / x) - (1.0 / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / x) - (1.0d0 / x)
end function

Java

public static double code(double x) {
	return (1.0 / x) - (1.0 / x);
}

Python

def code(x):
	return (1.0 / x) - (1.0 / x)

Julia

function code(x)
	return Float64(Float64(1.0 / x) - Float64(1.0 / x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / x) - (1.0 / x);
end

Wolfram

code[x_] := N[(N[(1.0 / x), $MachinePrecision] - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.7%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

3. Simplified 70.7%

   \[\leadsto \color{blue}{\frac{1}{1 + x} + \left(\frac{-2}{x} + \frac{1}{x + -1}\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 69.6%

   \[\leadsto \color{blue}{\frac{1}{x}} + \left(\frac{-2}{x} + \frac{1}{x + -1}\right) \]

6. Taylor expanded in x around inf 69.4%

   \[\leadsto \frac{1}{x} + \color{blue}{\frac{-1}{x}} \]

7. Final simplification 69.4%

   \[\leadsto \frac{1}{x} - \frac{1}{x} \]
8. Add Preprocessing

Alternative 10: 6.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 x))

C

double code(double x) {
	return 1.0 / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / x
end function

Java

public static double code(double x) {
	return 1.0 / x;
}

Python

def code(x):
	return 1.0 / x

Julia

function code(x)
	return Float64(1.0 / x)
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / x;
end

Wolfram

code[x_] := N[(1.0 / x), $MachinePrecision]

Derivation

1. Initial program 70.7%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

3. Simplified 70.7%

   \[\leadsto \color{blue}{\frac{1}{1 + x} + \left(\frac{-2}{x} + \frac{1}{x + -1}\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 69.6%

   \[\leadsto \color{blue}{\frac{1}{x}} + \left(\frac{-2}{x} + \frac{1}{x + -1}\right) \]

6. Taylor expanded in x around 0 5.3%

   \[\leadsto \color{blue}{\frac{-1}{x}} \]
7. Step-by-step derivation

   1. div-inv 5.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{1}{x}} \]

   2. pow1 5.3%

      \[\leadsto -1 \cdot \frac{1}{\color{blue}{{x}^{1}}} \]

   3. metadata-eval 5.3%

      \[\leadsto -1 \cdot \frac{1}{{x}^{\color{blue}{\left(\frac{2}{2}\right)}}} \]

   4. sqrt-pow1 57.5%

      \[\leadsto -1 \cdot \frac{1}{\color{blue}{\sqrt{{x}^{2}}}} \]

   5. pow2 57.5%

      \[\leadsto -1 \cdot \frac{1}{\sqrt{\color{blue}{x \cdot x}}} \]

   6. sqr-neg 57.5%

      \[\leadsto -1 \cdot \frac{1}{\sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}} \]

   7. sqrt-unprod 3.7%

      \[\leadsto -1 \cdot \frac{1}{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}} \]

   8. add-sqr-sqrt 6.6%

      \[\leadsto -1 \cdot \frac{1}{\color{blue}{-x}} \]

   9. metadata-eval 6.6%

      \[\leadsto -1 \cdot \frac{\color{blue}{--1}}{-x} \]

   10. frac-2neg 6.6%

       \[\leadsto -1 \cdot \color{blue}{\frac{-1}{x}} \]

8. Applied egg-rr 6.6%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1}{x}} \]
9. Step-by-step derivation

   1. associate-*r/ 6.6%

      \[\leadsto \color{blue}{\frac{-1 \cdot -1}{x}} \]

   2. metadata-eval 6.6%

      \[\leadsto \frac{\color{blue}{1}}{x} \]

10. Simplified 6.6%

    \[\leadsto \color{blue}{\frac{1}{x}} \]
11. Add Preprocessing

Alternative 11: 5.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ -1.0 x))

C

double code(double x) {
	return -1.0 / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-1.0d0) / x
end function

Java

public static double code(double x) {
	return -1.0 / x;
}

Python

def code(x):
	return -1.0 / x

Julia

function code(x)
	return Float64(-1.0 / x)
end

MATLAB

function tmp = code(x)
	tmp = -1.0 / x;
end

Wolfram

code[x_] := N[(-1.0 / x), $MachinePrecision]

Derivation

1. Initial program 70.7%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

3. Simplified 70.7%

   \[\leadsto \color{blue}{\frac{1}{1 + x} + \left(\frac{-2}{x} + \frac{1}{x + -1}\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 69.6%

   \[\leadsto \color{blue}{\frac{1}{x}} + \left(\frac{-2}{x} + \frac{1}{x + -1}\right) \]

6. Taylor expanded in x around 0 5.3%

   \[\leadsto \color{blue}{\frac{-1}{x}} \]
7. Add Preprocessing

Alternative 12: 5.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{-2}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ -2.0 x))

C

double code(double x) {
	return -2.0 / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-2.0d0) / x
end function

Java

public static double code(double x) {
	return -2.0 / x;
}

Python

def code(x):
	return -2.0 / x

Julia

function code(x)
	return Float64(-2.0 / x)
end

MATLAB

function tmp = code(x)
	tmp = -2.0 / x;
end

Wolfram

code[x_] := N[(-2.0 / x), $MachinePrecision]

Derivation

1. Initial program 70.7%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

3. Simplified 70.7%

   \[\leadsto \color{blue}{\frac{1}{1 + x} + \left(\frac{-2}{x} + \frac{1}{x + -1}\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 5.3%

   \[\leadsto \color{blue}{\frac{-2}{x}} \]
6. Add Preprocessing

Developer Target 1: 99.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{2}{x \cdot \left(x \cdot x - 1\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (* x (- (* x x) 1.0))))

C

double code(double x) {
	return 2.0 / (x * ((x * x) - 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / (x * ((x * x) - 1.0d0))
end function

Java

public static double code(double x) {
	return 2.0 / (x * ((x * x) - 1.0));
}

Python

def code(x):
	return 2.0 / (x * ((x * x) - 1.0))

Julia

function code(x)
	return Float64(2.0 / Float64(x * Float64(Float64(x * x) - 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / (x * ((x * x) - 1.0));
end

Wolfram

code[x_] := N[(2.0 / N[(x * N[(N[(x * x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024129 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

  :alt
  (! :herbie-platform default (/ 2 (* x (- (* x x) 1))))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))